L I B HAHY
OF THE
U N IVER_SITY
Of 1LLI NOIS
€21.365
Tje€55te
no. 40-^9
cop. 2
Digitized by the Internet Archive
in 2013
http://archive.org/details/evaluationofcros49macp
ANTENNA LABORATORY
Technical Report No. 49
EVALUATION OF CROSS-CORRELATION METHODS
IN THE UTILIZATION OF ANTENNA SYSTEMS
by
Robert H, MacPhie
25 January 1961
Contract AF33(616)-6079
Project No. 9-(13-6278) Task 40572
Sponsored by:
WRIGHT AIR DEVELOPMENT CENTER
Electrical Engineering Research Laboratory
Engineering Experiment Station
University of Illinois
Urbana, Illinois
ENGINEERING LIBKARY
ACKNOWLEDGMENT
The author is very grateful for the many helpful suggestions from
Professor Deschamps. The discussions with Professor Lo^ Mr. Craig Allen
and other members of the Antenna Laboratory were also of considerable' help,
ABSTRACT
The theory of the cross-correlation of signals of two or more antennas to
produce antenna product patterns is reviewed and presented in a generalized
form. The problem of distortion of amplitude modulation by the multiplicative
processes is considered and a system to remove this distortion by filtering
is proposed. An analysis is made of the correlation system output when in the
presence of a primary desired signal there is a secondary interfering signal
incident on the antenna system from a different direction. It is shown that
the rejection of this interference does not depend entirely on the product
pattern itself. Indeed^ if the interfering signals amplitude is relatively
low, the interference in the system output is determined mainly by a low-level
interference pattern which is essentially the sum of the factor patterns of
the individual antennas rather than their product. Some ostensibly good
product patterns have associated with them very poor low-level interference
patterns. The analysis considers the generalization to more than one inter-
fering signal and more than one pattern multiplication as well as the effect
of time averaging (filtering) which is sometimes possible, for example, in
the case of Radio Astronomy.
Use is made of the negative sidelobes of the product pattern to suppress
the integrated effect of average background noise. It is shown that for any
arbitrary noise background with the signal known to be absent the time average
of the total contribution of this noise in the system output can be reduced to
zero. Then when the signal is transmitted its time averaged effect alone will
be observed in the system output. It is assumed that the noise from any given
direction is a quasi-stationary random process which is statistically independent
of the noise from any other direction.
A discussion of the noise limitation on the formation of arbitrary
patterns from a two element array by means of multiplicative operations is
presented and it is shown that any background noise will generally cause a
shift in the apparent direction of arrival of the signal as 'seen" by the
two element product pattern. The formation of an arbitrary pattern from
this basic two element pattern with its error due to the shifting action
of the noise does nothing to reduce this error. Rather^ it only serves to
emphasize the apparent direction of arrival of the signal and if the noise
is relatively large the error between the apparent direction and the true
direction can be considerable.
CONTENTS
Page
1, I rr reduction 1
2, The Multiplication of Antenna Patterns bv Signal Cross-Correlation 3
2„1 Two Antenna Case 3
2„2 Demodulation and Multiplicative Distortion 5
2„3 N Antenna Case and Removal of Amplitude Modulation Distcr-
^ion by filtering 6
3, Analysis of the Effect of More Than One Plane Wave Incident en
the Antenna System 13
3 1 0;+put of An Additive System with Two Plane Waves Incident 13
3„2 Output of a Two Antenna Multiplicative System with Two
: nndpp' Plane Waves 14
3 3 Output of a Two Antenna Multiplicative System with R
Interfering Signals 22
3.4 '•.■*" erf erence in the Case of More Than One Multiplication 24
4c Suppr-? -" <~>n ft T he Effect of Average Background Noise by Use
of the Negative Sidelobes of the Product Pattern 28
5„ The NToise Lim ition on the Formation of Arbitrary Patterns
from a Two Element Interferometer by Means of Multiplicative
Operations 36
6, Cone 1 us i on' 42
Reference? 44
ILLUSTRATIONS
Figure
Number
Page
1. Mul tipli ca T j en of Two Antenna Patterns 4
2. Frequency Spectra of Signals at Successive Stages of the
Process of Removing Undesirable Modulation on the Signals
to be Cross-Correlated. 8
3. System for Removing Modulation from All but the Last
Product Signal for the Case of Multiplication of N
Antenna Patterns 10
4. Comparison of Product Pattern and Low-Level Interference
Pattern for the Compound Interferometer 19
5. System for the Suppression of the Average Value of the Output
of a Compound Interferometer Due to an Arbitrary Distribution of
Background Noise 34
6„ Phaser Representation of the Signal Plus Noise Output from a
Two Element Multiplicative Array. 40
7. <a) Interferometer Pattern of Signal Alone, (b) Interferometer
Pattern of Noise Alone, (c) Interferometer Pattern of Signal
Plus Noise^ (d) Synthesized Pattern of Signal Plus Noise 42
INTRODUCTION
For some time cross-correlation techniques have been used to obtain receiving
antenna patterns which are effectively the product of the patterns of two or
more different antennas. Practical examples of such systems are the Mill's
1 2
Cross and the Compound Interferometer both used in Radio Astronomy. In the
former, two long, mutually perpendicular antennas of lengths L and L in the
form of a cross, have their signals cross-correlated to produce a power pattern,
with a pencil beam, roughly equal to the voltage pattern of an antenna whose
area is L L , The Compound Interferometer places a uniformly weighted aperture
beside a simple interferometer and the cross-correlated output combines the
sharpness of the interferometer pattern with the directivity of the aperture.
If both are of length $ (hence the system is of length 2i ) , the product
power pattern is equal to the voltage pattern of a uniform aperture of length
4i.
In this report the product pattern for the general case of two antennas
separated by a distance i and with complex patterns is presented Then the
further generalization to the multiplication of N antenna patterns is made,
and a system to accomplish this is proposed. In this connection the problem of
demodulation is considered. The multiplicative operations cause distortion of
the modulating signal and it is shown that this multiplicative distortion can
be eliminated (in theory) by filtering.
The problem of rejecting interfering signals from arbitrary directions
other than that of the desired signal is more complicated than in the case of
additive arrays. It will be shown that the product pattern alone does not
describe fully the rejection of interference. Another pattern, which in general
depends on the relative directions of the primary and interfering signals, has
to be introduced. In the case of a single cross-correlation the pattern is
essentially the sum of the factor patterns of the two antennas rather than
their product It will be shown that some good product patterns have associated
with them rather poor low-level interference patterns „ More generally, if the
signals of N antennas are cross-correlated, in the system output there are N-l
distinct interference patterns all of which must be analysed when interference
rejection is required.
The product pattern, unlike the power pattern of additive antenna systems,
has negative sidelobes and in Section 4 it is shown that, they can be used to
eliminate the effect of background noise in the time-averaged output of the
correlation system. The proposed scheme will work only when it is known a
priori when the signal is present and when it is absent,, In addition the
noise from any direction must be quasi-stationary and statistically inde-
pendent of noise coming from any other direction„
Finally an analysis of the noise limitation on the formation of arbitrary
patterns from a two element array by means of multiplicative operations is
presented. It is shown that errors introduced by the noise into the two element
product pattern cannot be reduced in the formation from i t of patterns of
ostensibly much greater resolution.
2. THE MULTIPLICATION OF ANTENNA PATTERNS
BY SIGNAL CROSS -CORRELATION
2,1 Two Antenna Case
A diagram of the multiplication system for two antennas is shown in Figure
1. The signal from antenna A is
A(9,t) = Re A(0)e Jwt volts (1)
and that from antenna B is
B(6,t) = Re B(e)e" jf3isin6 e Jwt volts (2)
= Re B (Q) e JCOt volts (2a)
where 9 = angle which the propagation vector of an incident plane wave makes
with the normal to the line joining the two antennas,
t = time in seconds,
u> = angular frequency of the incident plane wave in radians per second,
i = spacing between the antennas in meters,
P = propagation constant in radians per meter,
Re indicates 'the real part of".
The pattern coefficients A(Q) and B (0) are in general complex. Note that the
effect of the spacing i between the antennas is the phase factor e " K
which is absorbed into B.(0)„
Now as is shown in Figure 1, the signal from antenna A enters a frequency
shifter (a nonlinear device) together with a local signal at frequency oj .
The output is filtered so as to pass only the difference frequency (oj = to - co )
LOCAL
PUMP
SIGNAL
(J
ANTENNA A
9
ANTENNA B
¥
W
i
> x
A(0,<U)
B^(0,W)
FREQUENCY
SHIFTER
AND
FILTER
A(0,o/)
It
MIXER
W.
i
Pp<0,W |f <)
SYNCHRONOUS
DETECTOR
I
R,(0,f)
'' '• Multiplication of Two Antenna Patt
erna
signal which is simply the signal from A shifted in frequency to co . Then this
signal and that from B enter a mixer which is a second nonlinear device
whose output contains among other components a signal at frequency to . By means
of a synchronous detector the amplitude of this signal can be obtained and it
is given by
p (e, i) = Re A(e) B*(e) o>
ft
where indicates the complex conjugate quantity. In trigonometric form, Equa-
tion (3) can be written as
p (e, i) = :A(e) B(e>l cos[pisine + a (e> - p(e)l (3a)
where a(G) = Arg A(Q) and (3(9) = Arg B(Q). Since P (Q,4) is the product of two
voltage patterns it varies as the power of the incoming waves and consequently
any comparison with additive antenna systems patterns will be with the power
patterns of those systems. We note incidentally from Equation (3) that if
/ = and A(Q) = B(Q), then the product pattern degenerates to an additive
systems power pattern given by
P o (0) = |A(e)^ 2 (4)
I
2.2 Demodulation and Multiplicative Distortion
I — .
If the incoming plane wave is modulated, with amplitude modulation for
example, then the output of the system will be
P 2 (0, F(t),l) =[14 mF(t)] 2 Re A(e) B*(e) (5)
= [1 + 2mF(t) + m F(t) ] Re A(e) B*(0) (5a)
where m = percent modulation, (0 < m < 1),
F(t) = modulating signal.
2 2
Hence there is square law distortion (m F(t) ) which can be significant for
large m. This problem also occurs in additive systems which employ square law
detection. Gray* has shown for the simple case of sinusoidal modulation that
for less than 10$ square law distortion m must be less than 40$ which is not
too severe a restriction. However, if N antenna patterns are multiplied together
(as will be shown in the next section) the output is proportional to
P N (G, F(t)) = [1 + m F(t)] N P (0) (6)
As N increases the restriction rapidly becomes intolerable; for N = 6 and 10$
2 2
distortion from only the m F(t) term in the binomial expansion, m must be
kept less than 8$, In the following section a system for multiplying N antenna
patterns which removes this multiplicative distortion is proposed.
2,3 N Antenna Case and Removal of Amplitude Modulation Distortion by Filtering
If there were N antennas the spectrum of the signal at the terminals of
each would be that of the carrier and its modulation while the amplitude and
phase of the signal would be determined by the antenna's pattern and its loca-
tion. Now since any one of the N antennas signals is modulated in the same way
as any other, nothing would be lost if the modulation were removed from the
carrier ol - • 1 1 but one of these signals. Not only will there be no loss of
information bill L1 Will be shown that this actually eliminates the multiplicative
distortion brouL'M aboul by the repeated cross-correlation of the antenna
'•8 .
1 Applied Electronics, Wiley, pp. 748-740
Let the N antennas have terminal voltages given by A (0, t,F(t)),
1
A (0,t,F(t)) A (9,t,F(t)) where
k N
A, (G,t,F(t)) = [1 + mF(t)] Re A (0) e "J Cpi k sin0-ct)
£ k k
= [1 + mF(t)] Re A (0) e'^ 1 (7)
k
For simplicity we consider the case of colinear antennas whose patterns are
functions of alone. The more general case of non-colinear antennas with
"two-dimensional" patterns (Q,§) is of no greater difficulty as far as the
multiplicative operations are concerned In Figure 2(a) are shown the spectra
of A (0, t, F( t)) and of the local signal at frequency co . The separation of
1
the local signals frequency co from the carrier's at to is A co and is chosen
soas to be slightly larger than the spectral width of one of the carrier's
sidebands. Hence, the output of the first frequency shifter (FS) will contain
the difference frequency at A co along with the modulation spectrum of the signal
This difference frequency can be held constant by an automatic frequency control
device (AFC). The low frequency output spectrum is shown in Figure 2(b). A
high Q filter tuned to frequency A co will remove the unwanted modulation on the
new "carrier" leaving a single frequency signal whose amplitude varies as the
antenna pattern A. (0), (see Figure 2(c)). The frequency shifting operation
1
to a much lower frequency is made necessary by the difficulty of removing the
modulation from an RF carrier by direct filtration.
A second frequency shifter (FS) ? this time with a local signal at frequency
U = co - 2 A co is used to return the unmodulated "carrier" at A co to the RF
SIGNALS INTO FIRST FREQUENCY SHIFTER
RADIO
(a)
FIRST PUMP
FREQUENCY . ... n/ ,. Q .
FREQUENCIES 4 ♦ A^.FM.e)
nrv .
J. cj ^^
(b)
.1 LOW FREQ. OUTPUT FROM FIRST FREQ. SH
♦ A{(A£J,F(6)),e) = A{«J,F(a»,G) SHIFTER DOWN
y<"~"^ /^N IN FREQUENCY TO AW
FTER
|A^(AW,e) /MODULATION SPECTRUM F(U)\
\REMOVED BY HIGH Q FILTER/
A6J
(d)
i SIGNALS INTO SECOND FREQ. SHIFTER
| A^(A6),9)
1
At)
2ND PUMP
FREQUENCY
(e)
T 2 2 Ad)
SIGNALS OUT OF 2ND FREQ. SHIFTER
A {| (6J-3A^J,e)
SUPPRESSED
CARRIER
A.(6J-A6J,0)
■li
ii SIGNALS INTO FIRST MIXER
(f)
CJ-3AO) Gi^ GO-ACJ O)
t A <
(g)
W-3A6)
OW MIXER
2
a., (6),F(6)),e
CJ-ACJ 0)
P 2 AlA^I' |P 2
i/ir\17ir\
A<0 2AOJ 3A6J
(h)
"P (ACJ.9) /MODULATION SPECTRUM F«J)
V REMOVED BY HIGH Q FILTER;
Figure 2.
FREQUENCY, RAD./SEC.
PROCESS CAN BE CONTINUED BY STARTING AT STAGE (d) ABOVE AND
USING A. {CJJ{Q),Q) IN THE NEXT MIXER OPERATION
BCy Spectra of Signals at Successive Stages of the Process of
Bmovlng f.'ndosi rable Modulation on the Signals to be Cross-Correlated,
range. The FS output will be a new carrier at frequency oo with sidebands at
oo - A oo and oo + A co. At this point it is necessary to suppress the carrier
(there is no technical difficulty in doing this), Then the two sidebands whose
amplitudes are proportional to A (Q) are mixed with the modulated signal from
1
antenna two, A (0, t,F(t)) as is shown in Figure 2(f). The low frequency
2
difference output spectra are shown in Figure 2(g). Again an AFC device is
used to hold this output constant in frequency. A filtertuned to A co passes
the product signal given by
P 2 (9,t, i 1 ,i 2 ) = Re A i (0) A i *(0) e
j Aw t
(8)
and which has all its modulation removed.
The process can be continued by returning this signal to RF as was done
in stages 2(d) and 2(e) (see Figure 2). Then by mixing with A ,(0, t,F(t)) and
filtering one obtains
P 3 (0,t,i 1 ,i 2 ,i 3 ) = Re A i (0) A i (0) A jf (0) e
-I *-i O
j A oo t
(9)
which again has all the modulation removed. A diagram of the system is shown
in Figure 3. The process is continued until the (N - 2)ND unmodulated pro-
duct is formed, i.e.,
N-2
j A w t
P N _ 1 (0,t,i i ,...i N _ 1 ) = Re
A A (0) A *(0)
4 x 2q-l 2q
A (0) e^
N-l
if N is even and > 3, and
(10)
Vl^^l'-W =R
N-l
n a ( Q ) a *(0)
4 2q-l 2q
j A oo t
if N is odd and > 2 r (in the above formula i is zero), Then, instead of
c
bo
•H
CO
O
3
T3
O
0,
W
3
J2
3
•H
■o
r (
a
a
•H
■^
bfl
H
e
3
•H
3
>
<H
e
o
0)
ad
a
h
Ed
U
>H
0)
a
d
u
+J
M
■
Fi
>)
03
<M
3
ho
•H
11
shifting this signal back up to RF, we mix it directly with the modulated signal
from the last antenna A (0, t,F(t)), The modulated RF difference frequency is
N
if the mixer output signal at this frequency is fed into a synchronous detec-
tor along with a reference signal also at co taken from the local source, then
the detector output will be
N
2
^(0, F(t) £ 2? l 3 ,...l }i ) = [1 + m F(t)] Re U A < 0) A
2q-l
(0) A, *(Q)
2q
(11)
if N is even and > 1 and
P N (0, F(t), i 2 ,i 3 , ...,1 N ) = [1 + m F(t)] Re
f N-l
q 2i A i (e) A % <°>
4 X 2q-1 *2q
A iN (8)
if N is odd and > 2.
The output has no multiplicative distortion and consequently m need not be
limited to say, less than 40$ as with a square-law detector. The practical
success of the system depends on the filtering efficiency which in turn depends
on the modulation spectrum. The noise problem should be alleviated by the
filtering but the extra multiplicative operations will clearly degrade the signal
to noise ratio. Finally, mention should be made of amplification which probably
will be necessary if N is large. The most convenient place for an amplifier
'would be immediately after the filter that removes the modulation from the
pseudo-carrier at A co. A narrow band low noise parametric amplifier could be
ised.
It should be noted that the pattern P ( 0, F . __ . , i_, . . , ,i ) is proportional
N (t) 2 ' N
;o the Nth power of the electric field strength of the incoming plane wave,
tence the equivalent "field strength pattern" (voltage) is
T> t Q V t +■ \
\ — ID (d T?f + \
e »
1
,N
l 1 CM
12
and the equivalent power pattern is
2
p N p (e,F(t)i 2 , ....i N ) = P N ( Q ,F(t),i 2 , ... i N ) N (is)
Any comparison with conventional additive patterns should be made on this
basis. In the rest of this report only the power pattern will be used and the
superscript will be omitted.
13
3. ANALYSIS OF THE EFFECT OF MORE THAN ONE PLANE
WAVE INCIDENT ON THE ANTENNA SYSTEM
In the preceding analysis, only a single plane wave incident at the arbitrary
angle 9 was considered. In this section it will be shown that when two plane
waves are incident at angles 9 and Q , the output of the synchronous detector
of the product system is in important respects different from the output of the
conventional square law detector of additive systems. We let one of the plane
waves of unit amplitude at an angle of incidence 9 be the desired signal. The
other, of complex amplitude Se \ and angle Q , is either due to a discrete,
remote, noise source or a secondary component of the desired signal which
arises when there is more than one transmission path from the source to the
receiver (multipath propagation).
3.1 Output of An Additive System wit h Two Plane Waves Incident
In an additive system the received terminal voltage before detection is a
linear superposition of the two signals
C(t;l,9;Se , %) = C(t;l,9) + C(t;Se ,S )
= Re]c(9) + Se'^1 CCQ^ \ e jwt (14)
The usual detector > a square law device, has a low frequency output proportional
to
. t , 2 2 2 .t
Ca^iSeW^ = !c(9)^ + S C(9j + 2S Re C(0; C(Q) e" j5i (1!
l ■ ' i i 1
)r alternatively
3
Welsby and Tucker have obtained some similar results in which they compare
multiplicative systems with additive systems employing linear rather than
square law detection.
14
I i ^l I 2 I I 2 2 1 I 2 ,
|cu,e;Se J , e 1 )| = |c(e)| + s |c(e 1 )| + 2s|c(e) c(e 1 )|cos l 1 -Ke)+4(e 1 )
(15a)
where £{Q) = ARG C(9) .
o
If the pattern is normalized with the primary signal incident at = i
the beam angle of the pattern, Equation (13a) can be written as
c(se J , 1 e 1 '>
= 1 + s
0(9^
+ 2s|c(e i )|cos(l 1 -t(0°)+^(e i )^ (16)
C(G 1 )
where c(9 ) = , The last two terms of Equation (16) represent interference
and its rejection is seen to depend on c(0 ) alone.
3.2 Output of a Two Antenna Multiplicative System with Two Incident Plane Waves
The output of the synchronous detector in the two antenna multiplicative
case and with two incident plane waves is naturally more involved than that of
the additive system. It is of the form
P 2 a,9;Se j ^ 1 9 1 ;i) = ReJA(e)B i *(e) + S 2 Af e^B^^
+ s(A(9)B i *(9 1 ) e ' l + A(0 l )B i *(9) e *
(17)
or alternatively (and with some rearranging)
,h
P (l,e;SB , 9,;i) = A(6) B(Q) cos r : p/sinn-:a(0.'-(3<0>
& 1
s a(Oj > B(e 1 )|cos lysine +a(e 1 )-p^e 1 )]
A(9) B(9)|cofl p;sinG-ta(0)-|i(9)] ■
A(Q i |
a(e 1 )-p(e)
l I n6 a(6 I -p(6)
B(9j )
bo)
cos| pislnSj -i a(0)-p(e,)]
i i no n(f)J |ii n '
(17a)
15
Now as in the additive case, if the pattern is normalized with the pri-
mary signal incident at 6 = 0° Equation (15a) becomes
P 9 (Se J , 1 Q t) = 1 + s 2 ia(9.)b(e i )
cos[ pisine Td(e )-p(e )]
~" c M[a(0° > ^(M o 5 TT " ~~
+ S ', ia(6, )
cos(e i ^a(e 1 )-p(o°))
II cos(a(0°)-p(0°))
+ ib(e,)
i J-
cos(pisin0 1 -^ 1 ^a(O°)-p(e i ))
cos(a(0O)-(3(0°))
(18)
J
A(e 1 )
B(e 1 )
where 0.(0^ = ^^J . b 'V = j^ •
Comparing Equations (16) and (18) we note that the basic difference is in the
last terms which is the major interference term when S <* 1. In the additive
case (Equation (16)) the rejection of interference depends on the voltage
pattern c(0 ): the worst case occurs when £ = C(O)-C(0.,) and the interference
1 11
term is 2S c(0 )| , In the multiplicative case (Equation (18.)) the low-level
interference is proportional, not to the product pattern itself but to the sum
of the normalized factor patterns | a(0 )| and lb(0 )| each of which is multi-
1 i 1
plied by a cosine term. Consequently the rejection of this first order ( - s)
interference by a multiplicative system cannot be determined by the product
pattern at all. It will be convenient to define the pattern associated with S
as the low-level interference pattern since it predominates when S is small and
& is negligible Hence we let
,ar0 )| cos(£ -q(0 1 )-(3(O°)) * IbO,)! cos(p.fsin0 -S.'crtcft-pfe))
cos(a(0°) - (3(0°))
This expression is clearly equal to or less than
ja(0 i )i -» jb(0 1 )
(19)
but the
2cos[ a(0°)-p(0°)]
exact nature of the pattern is quite complicated for this general case. Now
16
for any given system, i is fixed and the pattern is a function of the two
variables G-, and £ „ Likewise for any given Q there will be an £ which
ma
ximizes the magnitude of L(Q ,§ ,i). To determine this we differentiate with
respect to £ and equate the derivative to zero.
8 L<Q i ,£ v O 1
-laCe^l sin[^ 1 +a(e i )-p(0°)]+|b(e i )| sin(pisine i -S 1 +a(0 o )-p(6 1 ))
cos(a(0°)-p(0°))
=
(20)
from which we obtain the relation
la(0 )lsin(a(0 )-p(O°))-ib(0 )lsin (pfsin G.+a(O°)-p(0 ))
i- e i ,i; |" ra(e 1 )!cos(a(e 1 )-p(o°)T+ib(e 1 )lcos(piiin e 1 +a(o°>-p(e 1 ))
-p<e r
(21)
Now there are two values of £,(Q,i) (in the range [ 0, 27T]) that satisfy the above
equation, namely § (6,i) and £ (0 ,i) + 77. Substituting these values of £
into the expression for L(0 , 1, -O gives the worst possible patterns L(0 £ ,i )
and L(0 |.+ 7T, i), If the patterns a(0 ) and b(0 ) are real, Equation (21)
reduces to
€ j (0 I) ■ tan
b(9 ) sin (P i sin )
a(0 ) + b(0 ) cos (P 1 sin Gj)
(21a)
and the expr< lor the worst patterns are
'.'V V "
a(9 1 )+b(9 1 ) cos(Pisin Oj) ! 2 + | b(Q ] ) si n (Pi sin ) . '
a^Oj) + b(0 ) cosfPisin 9, )
(22)
cos (1 (9j , i ))
17
ue^ l I+ ir,i)
a(8,)tb(eJ cos(Pisin ) I 2 + J b(0 n ) sin(Pi sin n K 2
1 _1_ 1_ L _ 1 1 J
a(0 ) + b(0 ) cSsTPisin )
(22a)
cosf£ (0 I) + TT)
But cos (I (0 , I ) + 77) = -cos (| (0 i)) and hence
L(e x , Ij + ir,i) = -L(0 r e p I)
(23)
Subsequently we will consider only
L o (0 r i) = L(0 r e r i)
(24)
knowing that the alternative solution L(0 £, + 77 ? f ) is simply its negative.
If i = there is a further reduction to
L (0 ) = - [a(0 ) + b(0 )]
o 1 2 1 1
(22b)
Equation (22) can be put into a quite symmetric form by letting
M(0
, I) = a(0 ) + b(0 ) cos(P I sin )
(25)
and
N(0 r i) = b(0 ) sin(P I sin )
(26)
Then Equation (22) becomes
W U -2
M(e x , i) 2 + n(0 i) 2
cos
tan
.1 N( 6p i>
M(0 £)
(27)
18
We can therefore conclude that some apparently good product patterns
might have quite unacceptable low level interference patterns. As an example
of this we consider the Compound Interferometer mentioned in the introduction.
It consists of a uniformly weighted aperture of length i, adjacent to a simple
interferometer also of length i. A diagram of the system is shown in Figure 4,
The normalized factor patterns are
sin
P | sin 9 \
a(e 1 )
. _v 2
P i sin 0,
b(9 1 )
= cos
"'P i sin 9.
Hence the product pattern is proportional to
P I sin 0.
sm
p 2 ( 8i , D---PT
P i sin A
sin 9 n
cos
V
cos (P I sin )
sin
/'P £ sin fl
p 2 ,e 1 - i "' -tn
P i sin 9,
sin 9,
where x = — - —
sin(4x)
4x '
(28)
But the worsl l ->w level interference pattern is
L (x)
o 2
x 2 , , „ v 2
cos x cos 2x f (cos x sin 2x)
sin x
+ cos x cos 2x
, s i n x
. / • cos x cos 2x
tan ' -
c:ds x sin 2x
(28a)
19
O)
C/5
V
(
\
r
\
V
CT
<
I-
O
Q
O
cr
Q.
UJ
O
z.
Ul
cr
ui
Li-
ce:
UJ
UJ
> Z
uj or
5
o
. t=
4
)
0)
as
a,
0)
u
c
a>
i«
a>
0)
CD
>
M
o +j
•c
a
a) 0)
C w
CD -u
•o
c
3
o
o,
s
o
o
CD
o o
en <h
•r-l
h
IS
s
O
O
0)
3
CUD
•H
20
The two patterns are plotted in Figure 4 as functions of x. It should be
noted that this graph shows only the normalized patterns (each with its max-
imum equal to unity). In any particular case, the relative magnitudes of the
two patterns depend on the value of S. For example, suppose that S = 0.25
5
and corresponds to x = — 77. The output signal to interference ratio for
1 8
the worst case would be
R(0 1 S) = —
S P 2 (9 1 , 1) + 2S L o (0 p I)
(.25) (.127) + 2(.25) .395
s = .25
~ 577/8
00794 + .1974
(29)
= 4.88
It can be seen that the low-level interference pattern dominates in this case
because of its high sidelobes which are only 3 db down at x = +77, + 277 ( +377 etc.
2
Note that in some cases S P (0 , SL ) < and hence the worst case of
interference would occur when 2S L (0,, 1) were negative also. If L (0,, i)
o 1' o 1 '
is not negative as in the above example, we would select the alternate worst
pattern -L (0, , 1 ) = L (0 , , £ _ + 77 t ) (see Equation (23)). The worst possible
o 1 o 1' I
signal to interference ratio for the system, without regard to sign, is therefore
R <G., i, S
f> 1 ' '
)l =
S I P.O., 1)1 + 2S|L (0 i )
2 1' o 1 '
(30)
By way of comparison, the worst case for an additive system would be
R (6 , S)| ■
8 2 |c(9 1 )| 2 + 2810(8] »
(31)
21
In the latter equation it is seen that high directivity of c(0 ) also means
a large signal to interference ratio while the high "directivity'' of P (0 , I )
in the former equation can be negated by L (0 S.) to give a low net signal
to interference ratio.
It is important to realize^ however, that this situation obtains only
when the interfering signal is at its worst possible phase with respect to
the desired signal. Usually the interference phase £ is a time varying quantity
with the probability of any one value of £ (t) in the interval [0, 27T] being the
same as any other. Hence, the average of the low-level interference when taken
over a suitably long period of time approaches zero, i.e..
1 im
T -r
A(0 X )
Me)"
cos (P I sin + |(t) + a(Q ) - (3(e))
~cos (f3 £ "sin + cl(6) - (3(e)
(32)
P«9 , ) . cos ((3 i sin 0, - £(t) + a(0) - (3(0 ))
1
ft©;
1 r___ _
"cos Wl sin + o-(0) _r T(0))
dt =
In practice this means that a low pass filter would be used to reduce the
low-level interference signal in the system output. This filtering process
is quite feasible in Radio Astronomy (hence the usefulness of Compound Inter-
ferometer and indeed of the Mill's Cross), since only the steady state flux
of energy from each point on the celestial sphere is required. For instan-
taneous reception of modulated signals or for any application where a lengthy
time averaging is not practicable, the low-level interference pattern must
be considered.
22
3.3 Output of a Two Antenna Multiplicative System with R Interfering Signals
If in addition to the desired signal of unit amplitude incident at the
th
angle 0, there are R secondary interfering signals^ the q of which has
amplitude S e and direction , the output of the square law detector
q q'
of the additive system will be
j£.
j&,
6(1, S ie , i; ...; S R e , R )
R # -jb
= |'C(6)| + 2Re fo'e) 2 S C (0 )e q
q-1 q q
r r :
. Z SsSc(0)c(0)e
-, n q J" q r
q=l r=l
6 -I )
(33)
The analogous output from the cross-correlation system is
P 2 (l, 0, S x e e, .
ji f * R R
• • s o e R , D ^) = Re i A (e) B , (e) + s s s s
R ' R £ q r
q=l r=l
£(e > - l(e„)
A(9 ) B . (0 ) e q
q f r
R # -jl(9 )
r + A(0) 2 S B. (0 ) e
r i r
r=l
t-j€ e i
+ B. (0) Z S A(0 ) e
^ , q q
q=l
(34)
A comparison oJ » h<;se equations with those for a single interfering signal
shows thai th< ten again has the low-level interference pattern
van by Equation N9). The rejection of Interference Li
23
S . <1 for all q will depend primarily on this pattern which, as was shown
q
in Section 3.2, can be quite poor even when the product pattern itself is
good. The step from R discrete interfering signals to a continuous back-
ground of interference is quite straightforward, and the rejection of low-
level interference in this case also depends on the low-level interference
pattern.
A major hindrance in the design of a satisfactory interference pattern
is the interferometer factor cos (P i sin 0) due to the separation i of the
two antennas Its oscillations between +1 and -1 cause high sidelobes,
especially if bi0 ) has high sidelobes also.
Indeed, it is quite easy to show that it is impossible to get a product
pattern and a low-level interference pattern that both possess low sidelobes
if one of the factor patterns does not have them.
If I is quite large the cosine factor in the noise term
cos[P I sin + a(e ) - P(9 )]
s lace^ b( 0l )' Ho^To") -(3(o o )i
from Equation (18) will oscillate rapidly between -1 and +1 as varies
Hence, for a number of values of this noise term will be given by
!a(0 ) b(0 )1
S _ i y. sHa(0 ) b(0 )|
cos[a(0°) - P(0°)] L X
and if b(0 ) has high sidelobes for these values of the noise term will be
large. It can be reduced only by making ia(0 )| small, much smaller th
an
Ibce^
However, the low-level interference noise term is given by
24
cos (I + a(0 ) _ (3(0°)) cos (Pi sin - | * a(0°) - (3(0 ))
s ; a(0 ) __ _i 1 + |b(0 )| L_-i_ _ .J.
1 cos[a(0°) -P(0°)] cos;a(0 ) -(3(0°)]
As £ varies this noise component varies also. It can be shown that the max-
imum value is at least
S \ ibO.M - la(0 )| r
___^_ i i__i_ S - [b(0)| - |a(0 n )| '
cos[a(0°) - (3(0°)] [ 1 l
To minimize this component we therefore must make ia(0 )\ srf (b(0 , ) ■ ,
and this is not what is required for the suppression of the other noise compon-
ent. It follows that one or the other of the noise components can be minimized
but not both.
3.4 Interference in the Case of More Tnan One Multiplication
The case of the multiplication of N antenna patterns with R interfering
signals is obviously an involved situation. For N even the product pattern is
N/2 . jPl sin R j|
V 1 °< S l *, °! - *> - " A 2q -l (9)e * = V \q-l (9 j )
q=l v J=l
(35)
e jPi 2q-l Sin 9 j
-jPi .me r -jft -^ 2q .i Bin e k >
A 2q (6) e ^IS k e A 2q (0 k ) e
Bj way Oi b N i 4, R ■ 1, and let the patterns be real. Even with
the expression that results when put into trigonometric
25
P (1, 0: Se , . i i i ) = A (0)A(0)A(9)A (6)cos<Pl sine-Pi sinG+Pl sine)
1 * "7,
cos iPi sine -Pi sin0+Pi' sin0)
2 3 4
A (0 )
2 1
/A (0 )
S ! -r- 7 -r— cos(Pi sin0-Pi sin©
A ( e ) 2 3
1 1
+ Pi 4 sin0 + l 1 ) + a~7q7- cos(Pi 2 sin0 1 - Pi 3 sin0 4- 0| sine - ^)
A (0 )
t A / Q x ~ cos (Pi sin0 - Pi sin0 1 + Pi sine 4 ^)
W \
r-7r T - cos (Pi sine - Pi sine + Pi sine 1 - (L)
A (0) 2 3 41"
4
A (0 ) A (0 )
A (0) A 2 (0T- C ° S(Pi 2 5ine i " Pi 3 Sin6 + Pi 4 Sin9)
J. u
W A 3 (0 1 )
-nenrier cos((3i 2 sine - Pi 3 slne i + pi 4 sin9 - 2 V
J- o
VV A (6 )*
a (0TA-T0T cos((3 V inG " Pi 3 sine + p V in V
1 4
W A 3 ,0 1 )
1~(e) A (0) cos(Pi 2 sin 0i - Pi 3 sin0 1 + Pi^in©)
~K^Q) A 4 (0) cos (Pi ^^ - Pi 3 sin t Pi 4 sin ei f 2^)
VV A 4 ' e i ) I
^ 3 T0TA 4 l0T~' COS(,3i 2 Sin0 " P V ine ! + Pl 4 Sine i , |
26
[A (0 ) A (0 ) A (0 )
\m 1(8) v(e> —tfV-i " Pi 3 sine i " p V ine - V
A (G ) A (9 ) A (B )
V(8) 1(8) A*<9) c ° s(! V in9 - P V ln9 l - P V in6 l ' V
A (0 ) A (8 ) A (0 )
\ (9) 1(9) A* (9) c -«V ine i - ? V in6 * P V ln9 l " V
12 4
A (9 ) A (6 ) A (8 )
\ C 9 ) A (6) A*(9) <"»«V 1 '* 1 " P's'^! + "V 1 " 8 !
2 3 4
,!
, A (0 ) A (0 ) A (0 ) A (0 )
-4 I 1 1 2 1 3 1 4 1 ,n. . _
Tl) A o (0) A (9) A f8) " cos(Pi 2 Slne i
12 3 4
Pi sin0
+ Pi sine, )
4 1 I
(36)
2 3
There are no* 3 distinct interference patterns associated with S, S and S
4
in addition to the product pattern associated with S . It follows a fortiori
that care must be exercised in the design of product patterns involving severa]
multiplicative operations if interference is to be rejected.
It is important to note that even in the case where time averaging is
2
possible, there are still terms in the S interference pattern that are not
functions of £i t ) and hence cannot be averaged out. The terms of the S and
S patterns are all functions of £.(t) and can be removed. If 6 is and
the patterns ire normal ized, the time-averaged output reduces to
iyi, o° s, e t v i 3 , i> 4 ) = i + s 2 [■ 1 ce 1 ) a. 2 (e 1 ) coscPi^mo^
B) coB(Pl 4 «in6 1 ) • :i 2 U) \ ) '•/ ,J i ) cos (Pi si no - Pi sine J
27
* (6,) MM cos (-(3| sine, + Pi .sine.)
3141 31 41
+ S 4 a ce n ) a (6. ) a O-.) a (0 ) cos (Pi sine - Pi sin0 * Pi sine. ) (37)
in which we see that the S interference pattern is a sum of 4 cross products,
e.g., a (0 ) a (0 ) cos(Pi sin - Pi sin ), of the original patterns
^ J. *J3 1 ^ X *j X
modified as usual by a cosine term. A useful design would minimize this
pattern in the sidelobe region as well as the product pattern itself.
We note also that if N is odd the random phase angles from the signal as
well as the noise are present in all output terms and consequently time
averages of the product pattern systems for N odd all go to zero. Hence,
only an even number of antenna patterns can be cross-correlated in say a
Radio Astronomy application.
28
4. SUPPRESSION OF THE EFFECT OF AVERAGE BACKGROUND NOISE BY
USE OF THE NEGATIVE SIDELOBES OF THE PRODUCT PATTERN
An interesting and useful feature of product patterns is their negative
sidelobes. Thus they contrast with the power patterns of additive antennas
which are non-negative functions of x = Pi sin 8./2„ It will be shown that
for an arbitrary distribution of background noise in the absence of the signal
the pattern can be adjusted so as to reduce to zero the time-averaged response
to this noise in the system output. We assume that the noise coming from any
direction is a narrow band quasi-stationary random process, statistically
independent of the noise coming from any other direction. If its average
power density is given by S (x, t) (which is generally a slowly varying unkno\
function), then the time-averaged response of the additive antenna system
to this noise is
I (x , t) =
o'
S 2 (x, t)|C (x - x)! 2 dx (38)
7 o
2
where . C(x) " is the power pattern of the antenna with x corresponding to the
o
/ (3£ s i n 9 \ 2
main lobe direction I x = — . S (x } t) is clearly non-negative and
the integral will be positive for any noise background that is not identically
zero for all x.
For the produd pattern case, the time-averaged output is
I (x , t) =
J
r
S 2 (x, t) P o (x - x) dx (39)
and since P, (x) can tak'.- on negative values it should be possible to select
iny given time t the integral equation
29
r
S 2 (x, t ) P (x - x) dx = (40)
' o 2 o
To make Equation (40; hold for all t it is clear that P (x) must become a
slowly varying function of time in order to compensate for the variation
2
in S (x, t). Thus Equation (40) can be generalized to
r
S 2 (x, t) P (x - x, t) dx = (40a)
J 2 o
In addition, as the beam angle scans it will be necessary, in general, for the
pattern to change if the output due to the noise is to remain at zero.
In this derivation it is assumed that it is known a priori when the signal
is present and when it is absent. During the periods when it is absent the
antenna pattern is adjusted to eliminate the effect of the slowly varying
average background noise. Then when the signal is transmitted its effect
alone will be observed in the system's output, A slowly varying background
noise density will gradually change this ideal condition and so there should
be some prearranged sequence of time intervals in which the signal is turned
off and the receiver pattern modified to compensate for this gradual change
of background noise density.
To illustrate, let us consider the Compound Interferometer with the uni-
formly weighted aperture replaced by a linear array of m * 1 elements with
real and symmetric weighting (a = a where r indicates the r element to
r -r'
the right from the center of the array and -r indicates the r element to
the lef t; : The patterns of the two antennas are
m/2
A(x - x ) = 2_ a cos[2r(x - x )] (41)
o r=0 r *• o
Dl
B(x - x ) = cos[ m(x - x )]
o o
30
(42)
R« ■ n Pi sin
Pi sin 9 o
where x = - — — ■ — — , x = ■
2 ' o 2
and G is the beam angle of the patterns,
o
Now from Equation (3a) it follows that
m/2
P (x - x ) = 2 a cosi r(x - x )] cos[m(x - x )] cos[2m(x - x )]
2 o r o o o
r=0
(43)
m/2
m/2
= 2af(x-x)=2 af(x-x)
rr o rro
r=0 r=0
Substituting this into the expression for the output due to the arbitrary
background noise density (Equation (39)) we get
I(x , t) =
o'
m 2
= 2
r=0
2 m/2
S (x, t) 2 a f (x - x) d:
rro
r=0
S (x, t) f (x - x) dx
r o
(44)
and from Equation i40) we require that
m 2
2 a
, '
2 m/2
S x t) f fx - x) dx = 2 a q (x , t) =
r o r r o'
r=0
(45)
where q (x t )
r o
S (x, t) f (x - x ) dx
r o
(46)
To get the coil I , q (x , l) it is necessary to install switches in the
r o
trar. from the various array elements. If all switches but the
the center element' I Lne were open then the output from the system
i
31
I (x , t) = a q (x , t) (47)
o o o o o
and if during the measurement starting at, say, t , a is set equal to unity,
we get
q (x t ) = I (x t ) (47a)
o <r o o o o
Likewise opening; all but the lines from the a elements and setting a equal
ft. r r
to unity we get
q (x , t ) = I (x , t ) (48)
r o' o r o o
In this way all of the coefficients can be obtained.
We can think of the q (x , t)s as components of an — + 1 dimensional noise
r o- 2
vector Q'x t 1 whose scalar product with the weighting vector A('x , t ) also
o o o
f _ a. i dimensions, is required to be zero by Equation (45) „ It is clear
that there is no unique vector A(x , t ) that will satisfy this condition since
o o
any vector will do which lies in the hyperplane to which Q<x t ) is perpen-
o o
dicular Out of this infinity of vectors that satisfy Equation (45) it would
be best to select one which gave the maximum response to the signal when it
was incident at the beam angle 6 . This means that we pick Aix t ) such that
o o o
m/2
P (0) = 2 a (x , t ) = MAX (49)
2 r=0
However it is obvious that some limit on the size of the vectors A('x , t )
o o
themselves must be established because if any vector is doubled in size then
the sum of its components will also be doubled, Hence we seek out from all
32
vectors of a given length lying in the hyperplane the one whose sum of its
components is the largest. We set this length (without loss of generality)
equal to unity and we obtain the condition equation
m/2
2 a (x , t ) = 1 (50)
r=0 r ° °
Note that there are — + 1 coefficients a (x , t ) and still only 3
2 r o o
simultaneous equations. There is thus a possible — - 2 additional conditions
that could be satisfied, e.g. 1) first null occurs at 9 - A, 2) first
sidelobe has maximum at = TT , etc.
However, let us consider the five-element case, m = 4 in which there
are just three possible conditions; the three equations are
a (x , t ) + a, (x , t ) 4 a (x , t ) = MAX
o o' o loo 2 o 7 o
c (x , t ) a fx , t ) + c, (x , t ) a (x , t ) f c (x , t ) a <x , t ) -
oooooo lo'oloo 2 o o 2 o o
2 2 2
a (x , t ) + a, (x , t)+a„(x.t)=l
o o' o loo 2o'o
(51)
The last two equations represent a plane and a sphere, respectively, in three
dimensional space. Their intersection is a circle of unit radius centered at
the origin and a certain point (or points) on it satisfy the first equation.
To get this point I arrange' s method of undetermined multipliers can be used
and the following s< I -,\ equal ions results.
L + c (x , t ) X (x , t ) + 2a (x , t ) X (x , t ) »
-/ o o o O o o
33
1 .. c ix , t i X (x , t ) + 2a, (x . t ) \ (x t ) =
loo ooo loo lo o
(52)
1 - c (x , t ) v 'X , t ) + 2a (x , t ) X (x , t ) =
2 " o o ooo 2 o o loo
c (x , t ) a <x , t ) + c, (x , t ) a n (x , t ) + c„ (x t ) a (x t ) =
ooo o o o 1 o' o 1 o' o 2 o 7 o 2 o o
2 2 2
a ix t ) + a (x , t ) + a„ (x , t ) = 1
o o' o 1 o' o 2 o' o
A digital computer could be used to solve these equations for the weighting
coefficients a ( x s t ) a (x , t) and a (x , t ) Then instructions from
ooolo'o 2 o o
the computer could be fed back to each of the lines coming from the array
elements where a variable attenuator would be adjusted to give the correct
weighting. A diagram of the system is shown in Figure 5. Note that the
computer also sends phase shift instructions (for scanning) and switching
instructions i for obtaining the q (x , t ) coefficients) The conditions
r o o
fed into the computer are, of course, those that are mentioned above and
which give rise to Equations (52) which the computer must solve.
A final word of caution should be mentioned Although in theory the
equations can always be solved to reduce the noise output to zero, some noise
backgrounds can be eliminated only by resorting to a supergain condition For
example, a not uncommon case is that of a single point source of noise which
can be eliminated only if the pattern always has a null in the direction of
the point source This presents no difficulty until the beam angle x itself
approaches the angle of the point source. This requires the null to approach
the beam angle, and since the pattern is an analytic function the response
of the pattern to the signal incident at the beam angle will approach zero.
V
3
r-H
>
CO
bfl
cd
0)
>
<
CO
x:
+j
<H
O
c
o
•H
CO
W
a
<x
3
CO
(1)
XI
0J
in
•H
o
c
•o
c
3
o
bD
o
CQ
<H
o
c
o
•H
+J
3
X3
cd (-<
O -H
U
o
a) o
-t->
en co
>> A
CO -P
(D
3
be
•H
3 >>
a m
•m cd
3£
35
To maintain the response at, say, unit amplitude it will be necessary to
increase the magnitude of the weighting coefficients a (x , t ) which means
going to a supergain condition. Furthermore, if the beam angle and the angle
of the point source of noise coincide it will be necessary to split the beam to
satisfy Equation (45). This means that it is impossible to eliminate the noise
and not the signal if both are incident as plane waves at the same angle. How-
ever, there is usually more than one source of noise and, in theory at least,
the signal from a point source which may or may not coincide with one of the
noise sources can be retained and the noise rejected.
36
5. THE NOISE LIMITATION ON THE FORMATION OF ARBITRARY PATTERNS
FROM A TWO ELEMENT INTERFEROMETER BY
MEANS OF MULTIPLICATIVE OPERATIONS
5,6
Several writers •' have discussed the possibility of forming arbitrary
antenna patterns by multiplicative operations on the cross-correlated output
from a single pair of elements spaced a distance I apart^ i.e., from a simple
interferometer „ Patterns of arbitrary sharpness of the main lobe could be
synthesized and used to advantage in direction-finding applications. However^
no account was taken of noise in these schemes^ and it will be shown below
that noise severely limits their usefulness.
From Equation (3a) the pattern of the interferometer is
P (6, i) = cos (Pi sin 8)
(58)
It is raised to various powers from say to M and a linear combination of
these terms each weighted by a factor q is formed
M k
Q (x) = 2 q cos x
k=0
(59)
where x = Pi sin 9 But
k
cos X
k-1
k
1 l k
cos[ (k - 2r)x] -4 —
„ k k
2 la
(60)
if k is even, and
k
COS X
k-1
2 | ')
r,
cos[ (k - 2r) x]
(60a)
37
if k is odd and where
is the binomial coefficient. Hence
M
Q (x) = 2 w cos(kx)
k=0
where the w s are linear combinations of the q.'s. By a proper choice of the
k K
q :'s any arbitrary pattern of a 2M + 1 element array (M even) can be simu-
lated. If M is odd the array has 2M elements.
The assumption is made that there is no noise. In particular, it is
assumed that only a single plane wave ; the signal ; is incident on the antennas,
In the following it will be shown that if the signal is received in an
arbitrary background of noise there will generally be a shift in the apparent
direction of the signal as "seen" by the basic pattern cos (Pi sin 0) and the
process of forming a "better" pattern by multiplicative operations cannot
remove this error.
Let the signal of say unit average power be incident on the system from
direction x = Pi sin 9 and let the noise have an average power density given
2
by S (x) . Then the time-averaged output from the low-pass filter of the
correlation system is
I(x) =
o
;5( X;l - x) + S^( Xl ),
cos(x - x ) dx
oil
cos(x - x) +
o
S (x, ) cos x dx
1 11
cos X
S (x ) sin x dx
sin x
(61)
where x is the beam angle of the interferometer pattern. We see that the
38
terms in the square brackets are the Fourier coef f icients, a and b , of the
series expansion of S(x) given by
where
S(x) = Z a cos kx + 2 b sin k x
k=o k k=i k
cos(kx) cos(jx) dx = 6
sin(kx) sin(jx) dx = 6, ,
kj
sin(kx) cos(jx) dx = ,
and
5 kj =l, k-J
= 0, k £ j
Therefore Equation (60) can be expanded as
I < x ) = cos(x - x) + a cos x + b sin x
o o 1 o i o
= cosCx - x) + c cos(x - x )
o l o c
where
2 2 . 2 . -1
c, = a, + b, , x = tan
1 1 1 c
The arbitrary nod ie background is "seen" by the pattern as a point source at
39
ang
le x = Pi sin with average power c .
c c 1
Equation (63) can be reduced finally to
I(x ) = [ cos x + c cos x 1 cos x + [ sin x + c sin x ■] sin x
o 1 c J o 1 c o
d, cos(x - x )
1 o d
(66)
where
2 ,2
[cos x + c cos x + [sin x + c, sin x J
L l c J L 1 c J
x = tan
d
-1
sin x + c sin x
1 c_
cox x + c cos X
1 c.
(67)
The relations are best shown by the phasor diagram of Figure 6. The output is
jx
the scalar product of the unit scanning phasor e and the phasor sum e ' +
J x ^ J x h
c d
c e = de There is a magnitude error and an angular error in the esti-
mated value of the signal. However^ if x = x or x -77 the angular error is
ze
ro; the magnitude error is maximized and is - c. . Contrariwise if x =
1 c
x - cos (c ,/2) the amplitude of d is unity and no magnitude error occurs.
The maximum error in angle for a given c < 1 is (x -x) =- sin c
1 d max 1
and occurs at x = x + (77/2 + sin e, ) . If c ' • 1 the error can be as much
c 11 —
as 77,
The basic patterns due to the signal alone ; the noise alone and the signal
plus noise are shown in Figures 7a, 7b ? and 7c respectively. In Figure 7d a
pattern Q„(x - x ) synthesized from the basic signal plus noise pattern.
MOd or- r ,
d cos(x - x ) is shown. The error (x, - x ) occurring in the basic pattern
1 o d d
can in no way be reduced by this synthesizing of Q (x - x ) since the process
Mod
40
0)
a
SI.
in o
o
c
3 Z
M
H
^<
—i
0. in
o
<
3
x-a
— 1 r-l
(U
CO CU
c
05
a
bO
•^1
60 r-l
i-(
■H ffl
CO
z
CO C
bo
<M
VI
<H -H
O CO
c
c
C <H
u
u
U O
0)
4)
0)
4->
•P
P c
■p
■»->
P u
ed
0)
3) CO
0,
£
a, *>
p
k >-•
t<
in at
1 0)
0)
a) a,
•p
■u
p
0)
0)
d) T3
s
G
E a»
■s. o
«
V ^
l*
U -H
-^ 0>
CO
0) in
<H
<*-{
<H 0)
u
^
fc £
<D
0)
1) P
+)
P
P c
—^ a
G
c >>
J w
i—i
HH CO
d A u Q
D
O
"O
\
\\
tr
M,
UJ
CO
CO
<
1
1
z
I
w 1
X *J
^1
a:
\
0"
/ /
^
_l
z
CO
CO
O
CO
<
X
0.
\
1
\
\ \
AV
\
\ \
/ \
\
^
' /
\
^/
^
^ —
— \
X
\
\
z /
\
"O
\ 2
Z CO /
V
*— *•
\
< <\/
\
X\
1 7\.
\
1
\ X ^«- — ►
to Q- / '
^
Xy
w\S\
\
\
\ \ "°v
•-*
G
ct)
V
C
B
hfi
0)
-H
r-H
w
W
CO
O
JG
p
^
<H
m
O
a
c
h
>.
■H
m
c«
*->
^
fit
*->
U
P
D
<
C
D.
CO
♦^
<x>
3
>
t<
*->
a
<u
Cfl
<D
Ul
U
IX
•H
H
O
r-<
u
^.
D.
•H
'/i
U5
•(->
Tj
g
rH
x:
.-1
3
E
a
fl
'D
Cb
^
3,
41
only defines more sharply the value of x for the benefit of the observer.
The synthesized pattern is limited not by its own shape and sharpness (in
theory it can be made to approach a delta) but by the basic interferometer
pattern to which any combination of point source plus background noise
appears as another equivalent point source. This is all that the synthesized
pattern will ever "'see". If there is a fairly low signal to noise ratio it
can be concluded that a method of pattern syntheses such as this can lead
to large errors in the estimated value of the angle of incidence of the
signal .
42
6. CONCLUSIONS
It has been shown that the cross-correlation of antenna voltages to
obtain product patterns gives rise to two major problems:
1. Multiplicative distortion of amplitude modulation,
2. Interference patterns which differ from the original product pattern
and which can have high sidelobes even when the product pattern itself has
low sidelobes.
The first problem can in theory be overcome by a rather complicated
system which requires frequency shifting and filtering. An analysis of the
second problem shows that if instantaneous reception of, say, modulated signals
is required, special care must be exercised in the design of both product
and interference patterns if an adequate rejection of interference is to be
achieved. For systems in which the time average of the incoming signals is
all that is required the problem simplifies considerably but if N, the number
of patterns to be multiplied, is large it is still quite involved.
It has also been shown that product patterns possess a distinct advantage
over conventional additive power patterns in that their negative sidelobes can
be used to suppress the integrated effect of any arbitrary distribution of
background noise.
Finally, the artificial formation of arbitrarily sharp antenna patterns
by multiplicative operations on the time-averaged output of a two element
interferometer system has been shown to be of limited value in the presence
of ,i background noise distribution because the noise causes a shift in the
apparent direction of arrival of the signal. This shift is not reduced by
formation '<( i more directive" pattern whose effect is therefore nothing
43
more than to emphasize the apparent direction of arrival of the signal as
indicated by the interferometer pattern. If the noise is relatively strong
the error in the apparent direction of arrival can be quite large.
44
REFERENCES
1. Mills, B. Y. and Little, A. G., "A High-Resolution Aerial System of a
New Type." Austral. J. Phys. , Vol. 6, pp. 272-278, September, 1953.
2. Covington, A. E. and Broten, N. W "An Interferometer for Radio
Astronomy with a Single Lobed Radiation Pattern," IRE Trans, PGAP,
AP-5, No. 3, pp. 247-255, July, 1957.
3. Welsby, V, G. and Tucker, D. G. , "Multiplicative Receiving Arrays,"
J, Brit, IRE, 19, pp. 369-382, June, 1959
4. Berman, A. and Clay, C. S., "Theory of Time-Averaged Product Arrays,"
J. Acoustical Soc. America , 29, No. 7, pp. 805-812, August, 1957.
5. Drane, C. J,, Jr., "Phase Modulated Antennas," presented at IRE-URSI
Joint Fall Meeting, 1959, San Diego, California.
6. Brown, J. L M Jr. and Rowlands, R. 0., "Design of Directional Arrays,'
J. Acoustical Soc. America , 31, No. 12, pp. 1638-1643, December, 1959.
7. Dwight, H. B., Tables of Integrals and Other Mathematical Data ,
The Macmillan Company, p.l.
ANTENNA LABORATORY
TECHNICAL REPORTS AND MEMORANDA ISSUED
"Synthesis of Aperture Antennas, " Technical Report No : 1, C.T.A, Johnk,
October. 1954. *
A Synthesis Method for Broad-band Antenna Impedance Matching Networks,"
Technical Report No. 2, Nicholas Yaru, 1 February 1955
The Asymmetrically Excited Spherical Antenna," Technical Report No, 3,
Robert C Hansen. 30 April 1955,*
"Analysis of an Airborne Homing System," T ech nical Report No, 4, Paul E„
Mayes r 1 June 1955 ( CONFIDENTIAL) .
Coupling of Antenna Elements to a Circular Surface Waveguide, Technical
Report No 5, H. E King and R. H. DuHamel , 30 June 1955.*
Axial ly Excited Surface Wave Antennas," Technical Report No. 7, D, E. Royal,
10 October 1955 *
Homing Antennas for the F-86F Aircraft f 450-2500mc) , " Technical Report No. 8,
P.E t Mayes R.F, Hyneman, and R.C. Becker, 20 February 1957, (CONFIDENTIAL).
Ground Screen Pattern Range," Technical Memorandum No 1, Roger R. Trapp,
10 July 1955 * ~~~ ~~~
Conj rag t AF33 ( 61 6 i ^3220
Effective Permeability of Spheroidal Shells," Technical Report No. 9, E. J.
Scott and R H DuHamel, 16 April 1956.
An Analytical Study of Spaced Loop ADF Antenna Systems," Technical Report
No. 10, D G Berry and J B, Kreer, 10 May 1956.
"A Technique for Controlling the Radiation from Dielectric Rod Waveguides,"
Tech_nical_Repot No. 11, J W. Duncan and R = H DuHamel, 15 July 1956.*
Directional Characteristics of a U-Shaped Slot Antenna, ' Technical Report
No^_12, Richard C Becker, 30 September 1956 „**
Impedance of Ferrite Loop Antennas," Technical Report No 13, V, H, Rumsey
and W. L Weeks. 15 October 1956
"Closely Spaced Transverse Slots in Rectangular Waveguide," Technical Report
No^_l_4, Richard F Hyneman, 20 December 1956.
Distributed Coupling to Surface Wave Antennas," Technical Report No 15.
jj^ ' — — — *
Ralph Richard Hodges, Jr , 5 January 1957.
The Characteristic Impedance of the Fin Antenna of Infinite Length, Technical
Report No 16 Robert 1 Carrel, 15 January 1957
On the Estimation of Ferrite Loop Antenna Impedance, Technical Re port No. 17,
Walter L. Weeks, 10 April 1957.*
A. Note Concerning a Mechanical Scanning System for a Flush Mounted Line Source
Antenna fechnic .1 Report No. 18, Walter L. Weeks, 20 April 1957,
Broadband logarithmically Periodic Antenna Structures. Technical Report No . 19
R K DuHamel and D E. Isbell, 1 May 1957.
Frequency Independent Antennas," Tec hnical Report No 20 V. H Rumsey, 25
October 1957
The Equiangular Spiral Antenna," Techni cal Report No, 21, J D, Dyson, 15
September 195?
Experimental Investigation of the Conical Spiral Antenna Technical Report
Vo 22. R L. Carrel 25 May 1957.** ' """
Coupling between a Parallel Plate Waveguide and a Surface Waveguide," Technical
Report \ T o. 23 E ' Scott, 10 August 1957.
Launching Efficiency of Wires and Slots for a Dielectric Rod Waveguide,"
rechni .. | i 24 J. W. Duncan and R. H DuHamel, August 1957.
The Characteristic Impedance of an Infinite Biconical Antenna of Arbitrary
Cross Section, hnical Repo rt No. 25, Robert L Carrel, August 1957.
-Backed Slot Antennas, ' Technical R eport No. 26_, R. J. Tector, 30
October 1957
Coupled Waveguide Excitation of Traveling Wave Slot Antennas/ Techni cal
Re] [0 27 W I Weeks, 1 December 1957.
Pti Red ingular Waveguide Partially Filled with Dielectric,"
28 W I Weeks, 20 December 1957
p< r lint Length of Biconical Structures of Arbitrary
Cross rechnli i] Rep ort No. 29 , J. D Dyson. 10 January 1958
Lcally Periodic Antenna Structur Technical Report No. 30,
D ^58
( ' pular Slots,' technical Reporl ^'< 31, N. J.
; , L958
tation oi i Surface Wave on a Dielectric Cylinder/'
I w Du J5 May L058
"A Unidirectional Equiangular Spiral Antenna, " Technical Report No. 33, J. D.
Dyson, 10 July 1958
Dielectric Coated Spheroidal Radiators," Technical Report No. 34^ W. L Weeks,
12 September 1958.
"A Theoretical Study of the Equiangular Spiral Antenna," Technical Report
No. 35, P E Mast, 12 September 1958.
Contract AF33 '616) -6079
Use of Coupled Waveguides in a Traveling Wave Scanning Antenna,' Technical
ILERELLJ^JL 6 -.' R H MacPhie, 30 April 1959.
On the Solution of a Class of Wiener-Hopf Integral Equations in Finite and
Infinite Ranges, Technical Report No. 37, Raj Mittra, 15 May 1959.
"Prolate Spheroidal Wave Functions for Electromagnetic Theory," Technical
Repojr£_No ^_38 , W, L, Weeks, 5 June 1959.
Log Periodic Dipole Arrays," Technical Report No. 39, D E Isbell, 1 June 1959.
A Study of the Coma-Corrected Zoned Mirror by Diffraction Theory," Technical
Report No^ 40. 3. Dasgupta and Y. T„ Lo, 17 July 1959
The Radiation Pattern of a Dipole on a Finite Dielectric Sheet/" T echnical
ReportNq 41 KG Balmain, 1 August 1959
The Finite Range Wiener-Hopf Integral Equation and a Boundary Value Problem
in a Waveguide. Technical Report No. 42, Raj Mittra, 1 October 1959.
"impedance Properties of Complementary Mul titerminal Planar Structures "
Te chjrica.1 JRepor t No __43_, G, A, Deschamps, 11 November 1959
On the Synthesis of Strip Sources," Technical Report No 44 Raj Mittra,
4 December 1959
Numerical Analysis of the Eigenvalue Problem of Waves in Cylindrical Waveguides,"
Technical Report No^ 45. C H Tang and Y. T, Lo, 11 March I960,
Ve* Circularly Polarized Frequency Independent Antennas With Conical Beam or
Omnidirectional Patterns," Tec hnica l Report No, 46, J.D Dyson and P.E. Mayes,
20 June 1960
Logarithmically Periodic Resonant-V Arrays,'" Technical Report No, 47, P.E.
Waves and R L Carrel, 15 July 1960.
* Copies available for a three week loan period
Copies no longer available.
AF 33(616) -6079
DISTRIBUTION LIST
One copy each unless otherwise indicated
Commander
Wright Air Development Center
Attn: WCOSI, Library
Wright-Patterson Air Force Base, Ohio
Commander
U.S. Naval Air Test Center
Attn: ET-315, Antenna Section
Patuxent River, Maryland
Chief
Bureau Naval Weapons
Department of the Navy
Attn: (RR-13)
Washington 25, D.C.
Commander
USA White Sands Signal Agency
White Sands Proving Command
Attn: SIGWS-FC-02
White Sands, New Mexico
Director
Air University Library
Attn: AUL-8489
Maxwell AFB, Alabama
Army Rocket and Guided Missile Agency
U.S. Army Ordnance Missile Agency
Attn: ORDXR-OMR
Redstone Arsenal, Alabama
Commander
Air Force Missile Test Center
Attn: Technical Library
Patrick Air Force Base, Florida
Commander
Aero Space Technical Intelligence Center
Attn: AFCIN-4c3b, Mr. Lee Roy Hay
Wright-Patterson AFB, Ohio
Director
Ballistics Research Lab.
Attn: Ballistics Measurement Lab.
Aberdeen Proving Ground, Maryland
Commander
801st Air Division (SAC)
Attn: DCTT, Major Witry
Lockbourne Air Force Base, Ohio
Office of the Chief Signal Officer
Attn; SIGNET-5
Eng , & Technical Division
Washington 25, D.C.
Director
Air University Library
Attn: AUL-9642
Maxwell Air Force Base, Alabama
National Bureau of Standards
Department of Commerce
Attn; Dr, A, G. McNish
Washington 25, D.C,
Director
U.S. Navy Electronics Lab.
Point Loma
San Diego 52, California
*Wright Air Development Division
Attn: N,C, Draganjac, WWDBEG
Wright-Patterson AFB, Ohio (2 copies)
Chief
Bureau of Aeronautics
Attn; Aer-EL-931
Department of the Navy
Washington 25, D.C,
Armed Services Technical Information Agency
ATTN: TIP-DR
Arlington Hall Station
Arlington 12, Virginia (10 copies)
(Excluding Top Secret and Restricted
Data) (Reference AFR 205-43)
AF 33(616^-6079
Commander
Wright Air Development Center
Attn: F, Behrens, WCLKR
Wright-Patterson AFB, Ohio
Commander
Air Research & Development Command
Attn RDTC
Andrews Air Force Base
Washington 25. D,C .
Director
Naval Research Laboratory
Attn: Dr , A, Marston
Anacostia
Washington 25, D C
Director, Surveillance Dept „
Evans Area
Attn; Technical Document Center
Belmar, New Jersey
Commander
Hq , Air Force Cambridge Research Center
ATTN CRRD, C, Sletten
Laurence G Hanscom Field
Bedford, Massachusetts
Commander
Air Proving Ground Command
At f n Classified Technical Data
Branch D 01
Eglin Air Force Base, Florida
Direc tor
Research and Development Command
Hq I SAF
Attn AFDRD-RF
Washington 25, D C
Commander
Air Force Ballistics Missile Division
Attn Technical Library
Air Force Unit Post Office
Los Angeles 45, California
Commander
Air Force Missile Development Center
Attn Technical Library
Holloman Air Force Base, New Mexico
Commander
801 st Ai r Divl Bion ( SA< '
Attn DCTTTD, Ma jor Hougan
Lockbourne Air Force Base, Ohio
Commander
Rome Air Development Center
At t n i>, ii>
Qrif fla Air Forc< Base, New York
Chief, Bureau of Ships
Department of the Navy
Attn; Code 838D
Washington 25, D.C ,
Commanding Officer & Director
U.S. Navy Electronics Laboratory
Attn; Library
San Diego 52, California
Andrew Alford Consulting Engineers
Attn. Dr A, Alford
M F Contract AF33( 600)-36l08
299 Atlantic Avenue
Boston 10, Massachusetts
ATA Corporation
1200 Duke Street
Alexandria, Virginia
Bell Telephone Labs , Inc
Attn: R, L. Mattingly
M F Contract AF33(, 616) -5499
Whippany, New Jersey
Bendix Radio Division
Bendix Aviation Corporation
Attn, Dr, K. F, Umpleby
M [• (Onttact AF33( 600) -35407
Towson 4, Maryland
Boeing Airplane Company
Attn C , Arms! rong
M/F Contract AF33( 600)-36319
7755 Marginal Way
Seattle, Washington
Doc i ng Ai rpl am C> irnpany
Attn Robert Shannon
W I Contrad AF33(600)-35992
Wichita, Ka nsas
AF 33(616)-6079
Canoga Corporation
M/F Contract AF08( 603) -4327
5955 Sepulveda Boulevard
P.O. Box 550
Van Nuys, California
Dome & Margolin, Inc.
M/F Contract AF33(600)-35992
30 Sylvester Street
Westbury
Long Island, New York
Dr. C. H. Papas
Department of Electrical Engineering
California Institute of Technology
Pasadena, California
Chance-Vought Aircraft Division
United Aircraft Corporation
Attn: R.C. Blaylock
THRU: BuAer Representative
M/F Contract NOa(s) 57-187
Dallas, Texas
Collins Radio Company
Attn: Dr. R. H. DuHamel
M/F Contract AF33(600)-37559
Cedar Rapids, Iowa
Douglas Aircraft Co., Inc.
Attn: G. O'Rilley
M/F Contract AF33(600)-25669 &
AF33(600)-28368
Tulsa, Oklahoma
Exchange and Gift Division
The Library of Congress
Washington 25, D.C . (2 copies)
Fairchild Engine & Airplane Corp.
Fairchild Aircraft Division
Attn: Engineering Library
S. Rolfe Gregory
M/F Contract AF33(038)-18499
Hagerstown, Maryland
CONVAIR
Attn: R. Honer
M/F Contract AF33(600)-26530
San Diego Division
San Diego 12, California
Dr. Frank Fu Fang
Boeing Airplane Company
Transport Division, Physical Research
Renton, Washington
General Electric Company
Attn: D, H. Kuhn, Electronics Lab,
M/F Contract AF30(635)-12720
Building 3, Room 301
CONVAIR
Fort Worth Division
Attn: C, R. Curnutt
M/F Contract AF33(600)-32841 & College Park
AF33(600)-31625 113 S. Salina Street
Fort Worth, Texas Syracuse, New York
Department of Electrical Engineering
Attn; Dr, H, G, Booker
Cornell University
Ithaca, New York
University of Denver
Denver Research Institute
University Park
Denver 10, Colorado
Dalmo Victor Company
Attn: Engineering Technical Library
M/F Contract AF33( 600)-27570
1515 Industrial Way
Belmont, California
General Electronic Laboratories, Inc.
Attn: F. Parisi
M/F Contract AF33(600)-35796
18 Ames Street
Cambridge 42, Massachusetts
Goodyear Aircraft Corporation
Attn: G. Welch
M/F Contract AF33(616)-5017
Akron, Ohio
Granger Associates
M/F Contract AF19(604)-5 509
966 Commercial Street
Palo Alto, California
AF 33(616)-6079
Grumman Aircraft Engineering Corp.
Attn J, S, Erickson
Asst , Chief, Avionics Dept
M F Contract NOa(s) 51-118
Bethpage, Long Island, New York
ITT Laboratories
Attn: A. Kandoian
M/F Contract AF33(616)-5120
500 Washington Avenue
Nutley 10, New Jersey
Gulton Industries, Inc,
Attn B Bittner
M F Contract AF33(600) -36869
P.O Box 8345
15000 Central, East
Albuquerque, New Mexico
Hallicraf ters Corporation
Attn D. Herling
M F Contract AF33(604)-21260
440 W, Fifth Avenue
Chicago, Illinois
Technical Reports Collection
Attn, Mrs E. L, Hufschmidt
Librarian
303 A Pierce Hall
Harvard Iniversity
Cambridge 38, Massachusetts
Hoffman Laboratories, Inc.
Attn S. Varian (for Classified)
Technical Library (for
I nclassi f ied)
M F Contract AF33( 604) -1 7231
Los Angeles, California
Dr. R. F Hyneman
P Box 2097
Mail Stat ion C-152
Building 600
Hughes Ground Systems Group
Fullerton, California
HRB-Singer, I ne
Attn Mr R A Evans
Science Park
State College. I
Mr Dwighl I Bbe] i
4620 Sunnyside
Seal t : <■ 3, WashJ rig I on
ITT Laboratories
Attn;, L„ DeRosa
M F Contract AF33( 616)-5120
500 Washington Avenue
Nutley 10, New Jersey
ITT Laboratories
A Div. of Int, Tel. & Tel, Corp,
Attn: G B S. Giffin, ECM Lab,
3700 E. Pontiac Street
Fort Wayne, Indiana
Jansky and Bailey, Inc
Engineering Building
Attn: Mr. D. C, Ports
1339 Wisconsin Avenue, N.W.
Washington, D.C
Jasik Laboratories, Inc,
100 Shames Drive
Westbury, New York
John Hopkins University
Radiation Laboratory
Attm Librarian
M/F Contract AF33(616)-68
1315 St. Paul Street
Baltimore 2, Maryland
Applied Physics Laboratory
Johns Hopkins University
8621 Georgia Avenue
Silver Spring, Maryland
Lincoln Laboratories
Attn.. Document Room
M/F Contract AF19(122)-458
Massachusetts Institute of Technology
P.O. Box 73
Lexington 73, Massachusetts
AF 33(616)-6079
Litton Industries, Inc.
Maryland Division
Attn Engineering Library
M F Contract AF33( 600)-37292
4900 Calvert Road
College Park, Maryland
University of Michigan
Aeronautical Research Center
Attn: Dr, K, Seigel
M/F Contract AF30( 602)-1853
Willow Run Airport
Ypsilanti, Michigan
Lockheed Aircraft Corporation
Attn: C, D, Johnson
M/F Contract NOa(s) 55-172
P.O. Box 55
Burbank, California
Microwave Radiation Co,, Inc,
Attn: Dr, M, J. Ehrlich
M F Contract AF33(616'>-6528
19223 S. Hamilton Street
Gardena, California
Lockheed Missiles & Space Division
(Attn: E, A, Blasi
M/F Contract AF33( 600) -28692 &
AF33(616)-6022
Department 58-15
Plant 1, Building 130
Sunnyvale, California
The Martin Company
Attn; W, A, Kee, Chief Librarian
M/F Contract AF33(600)-37705
Library & Document Section
Baltimore 3, Maryland
Ennis Kuhlman
McDonnell Aircraft
P.O. Box 516
Lambert Municipal Airport
St, Louis 21, Missouri
Melpar, Inc .
Attn, Technical Library
M/F Contract AF19(604)-4988
Antenna Laboratory
3000 Arlington Blvd.
Falls Church, Virginia
Melville Laboratories
Walt Whitman Road
Melville, Long Island,
New York
Motorola, Inc ,
Attn: R, C, Huntington
8201 E, McDowell Road
Phoenix, Arizona
Physical Science Lab.
Attn. R. Dressel
New Mexico College of A and MA
State College, New Mexico
North American Aviation, Inc ,
Attn: J. D Leonard, Eng . Dept .
M/F Contract NOa(s) 54-323
4300 E, Fifth Avenue
Columbus, Ohio
North American Aviation, Inc,
Attn: H, A. Storms
M/F Contract AF33( 600)- 36599
Department 56
International Airport
Los Angeles 45, California
Northrop Aircraft, Inc.
Attn: Northrop Library, Dept. 2135
M F Contract AF33 (600)-27679
Hawthorne, California
Dr, R. E, Beam
Microwave Laboratory
Northwestern University
Evanston, Illinois
AF 33' 616'. -6079
Ohio State University Research
Foundation
Attn: Dr, T, C, Tice
M F Contract AF33(616)-6211
1314 Kinnear Road
Columbus 8, Ohio
University of Oklahoma Res, Inst.
Attn Prof, C, L, Farrar
M F Contract AF33( 616)-5490
Norman, Oklahoma
Dr. D. E. Royal
Ramo-Wooldridge, a division of Thompson
Ramo Wooldridge Inc,
8433 Fallbrook Avenue
Canoga Park, California
Rand Corporation
Attn: Librarian
M/F Contract AF18(600)-1600
1700 Main Street
Santa Monica, California
Philco Corporation
Government and Industrial Division
Attn Dr, Koehler
M. F Contract AF33(616)-5325
4700 Wissachickon Avenue
Philadelphia 44, Pennsylvania
Prof, A A, Oliner
Microwave Research Institute
Polytechnic Institute of Brooklyn
55 Johnson Street - Third Floor
Brooklyn, New York
Radiation, Inc ,
Technical Library Section
Attn Antenna Department
M F Contract AF33(600)-36705
Melbourne, Florida
Radio Corporation of America
RCA Laboratories Division
Attn Librarian
M F Contract AF33( 616)-3920
Princeton, New Jersey
Radioplane Company
M/F Contract AF33( 600) -23893
Van Nuys, California
Ramo-Wooldridge, a division of
Thompson Ramo Wooldridge, Inc.
Attn technical Information Services
8433 Fallbrook Avenue
P 0. Box 1006
Canoga Park California
Rantec Corporation
Attn: R. Krausz
M/F Contract AF19( 604)-3467
Calabasas, California
Raytheon Manufacturing Corp „
Attn; Dr, R, Borts
M/F Contract AF33(604)-15634
Wayland, Massachusetts
Republic Aviation Corporation
Attn: Engineering Library
M/F Contract AF33( 600)-34752
Farmingdale
Long Island, New York
Republic Aviation Corporation
Guided Missiles Division
Attn: J. Shea
M/F Contract AF33(616)-5925
223 Jericho Turnpike
Mineola, Long Island, New York
Sanders Associates, Inc.
95 Canal Street
Attn: Technical Library
Nashua, New Hampshire
Smyth Research Associates
Attn.; J. B. Smyth
3555 Aero Court
San Diego 11, California
Space Technology Labs, Inc,
Attn; Dr., R. C Hansen
P.O. Box 95001
Los Angeles 45, California
M/F Contract AF04( 647 ) -361
AF 33(616^-6079
Sperrv Gyroscope Company
Attn B, Berkowitz
M. F Contract AF33( 600)-28107
Great Neck
Long Island, New York
Stanford Electronics Laboratory
Attn. Applied Electronics Lab.
Document Library
Stanford Iniversity
Stanford, California
Stanford Research Institute
Attn; Mary Lou Fields, Acquisitions
Documents Center
Menlo Park, California
Stanford Research Institute
Aircraft Radiation Systems Lab,
Attn D, Scheuch
M F Contract AF33(616)-5584
Menlo Park, California
Sylvania Electric Products, Inc.
Electronic Defense Laboratory
M/F Contract DA 36-039-SC-75012
P.0, Box 205
Mountain View, California
Mr, Roger Battie
Supervisor, Technical Liaison
Sylvania Electric Products, Inc,
Electronic Systems Division
P.O. Box 188
Mountain View, California
Sylvania Electric Products, Inc ,
Electric Systems Division
Attn- C. Faflick
M/F Contract AF33(038)-21250
100 First Street
Waltham 54, Massachusetts
Technical Research Group
M/F Contract AF33< 61 6 ) -6093
2 Aerial Way
Syosset, New York
Temco Aircraft Corporation
Attn: G, Cramer
M/F Contract AF33( 600) -36145
Garland, Texas
Electrical Engineering Res, Lab,
University of Texas
Box 8026, University Station
Austin, Texas
A, S 5 Thomas, Inc„
M/F Contract AF04C 645;-30
161 Devonshire Street
Boston 10, Massachusetts
Westinghouse Electric Corporation
Air Arm Division
Attn: P, D, Newhcuser
Development Engineering
M/F Contract AF33(600)-27852
Friendship Airport
Baltimore, Maryland
Professor Morris Kline
Institute of Mathematical Sciences
New York University
25 Waverly Place
New York 3, New York
Dr , S. Dasgupta
Government Engineering College
Jabalpur, M.P,
India
Dr , Richard C. Becker
10829 Berkshire
Westchester, Illinois
Tamar Electronics, Inc,
Attn LB McMurren
2045 W Rosecrans Avenue
Gardena, California
The Engineering Library
Princeton University
Princeton, New Jersey
AF 33(616)-6079
Dr. B, Chatterjee
Communication Engineering Dept ,
Indian Institute of Technology
Kharagpur (S.E. Rly,)
India
Sperry Phoenix Company
Attn: Technical Librarian
P.O. Box 2529
21111 North 19th Avenue
Phoenix, Arizonia
Dr. Harry Letaw, Jr„
Raytheon Company
Surface Radar and Navigation Operations
State Road West
Wayland, Massachusetts
TVNews
OpenLibrary